View source for Sturm-Liouville operator

A [[Self-adjoint operator|self-adjoint operator]] generated by a differential expression

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and suitable boundary conditions in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907603.png" /> is a finite or infinite interval, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907604.png" /> are continuous real functions, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907605.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907606.png" /> (sometimes any operator generated by a quasi-differential expression analogous to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907607.png" /> is called so). Starting in 1830, J.Ch. Sturm and J. Liouville published a number of fundamental studies on the theory of the [[Sturm–Liouville problem|Sturm–Liouville problem]] on a finite interval.

A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907608.png" /> is called a regular end-point if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s0907609.png" /> is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076011.png" />. Otherwise this point is called a singular end-point. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076012.png" /> is called regular or singular depending on whether both end-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076013.png" /> are regular or not.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076014.png" /> be the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076015.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076016.png" /> is absolutely continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076017.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076018.png" /> be the subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076019.png" /> consisting of the functions with compact support. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076021.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076022.png" /> be the closure of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076024.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076025.png" /> is a symmetric operator, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076026.png" />. A Sturm–Liouville operator is an extension (restriction) of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076027.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076028.png" />.

1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076029.png" /> be regular, let the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076031.png" />, be linearly independent and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Then the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076033.png" /> that satisfy the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076035.png" />, is the domain of definition of some Sturm–Liouville operator. Conversely, the domain of definition of any Sturm–Liouville operator can be determined in this way.

Among the boundary conditions, an important place is occupied by the separated boundary conditions (or boundary conditions of Sturm type):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

and the mixed boundary conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076039.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076040.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076041.png" /> the conditions (5) are called periodic, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076042.png" /> anti-periodic (or semi-periodic).

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076043.png" /> be singular. The case when both end-points are singular can be reduced to the case of one singular end-point by splitting.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076044.png" />) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076045.png" /> be regular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076046.png" /> be singular, and let the number of independent solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076047.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076048.png" /> be equal to 1. Then the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076049.png" /> is said to belong to the case of a Weyl limit point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076050.png" />. The domain of definition of the Sturm–Liouville operator is determined by the boundary condition (3).

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076051.png" />) If the number of linearly independent solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076052.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076053.png" /> is 2, then the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076054.png" /> is said to belong to the case of a Weyl limit circle at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076055.png" />. The deficiency indices of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076056.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076057.png" /> in this case. The domain of definition of a Sturm–Liouville operator is described similarly to 1), replacing conditions (2) as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076058.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076061.png" /> are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076063.png" />, respectively, where

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here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076065.png" /> is the [[Wronskian|Wronskian]] of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076067.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076070.png" />, are the solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076071.png" /> with the initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076073.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076074.png" /> are the Kronecker symbols.

The resolvent kernel of a Sturm–Liouville operator is a [[Carleman kernel|Carleman kernel]]; moreover, the resolvent in cases 1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076075.png" />) is a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]], but in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076076.png" />) this is not necessarily the case.

The spectral expansion of a Sturm–Liouville operator in the case of a discrete spectrum (for example, in 1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076077.png" />)) is similar to the Fourier expansion in eigenfunctions of the Sturm–Liouville problem, and in the other cases it contains eigenfunctions that are not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076078.png" />.

Problems of finding conditions on the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076080.png" /> under which the Sturm–Liouville operator would have a discrete spectrum, or fills the whole line, and under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076081.png" /> would be of limit-point or limit-circle type, are of great interest. Completely general necessary and sufficient conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076083.png" />, which ensure that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076084.png" /> belongs to the limit-circle or limit-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090760/s09076085.png" /> type, are unknown (1984).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,   "Lineare Differentialoperatoren" , Akademie Verlag  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Akhiezer,   I.M. Glazman,   "Theory of linear operators in Hilbert space" , '''2''' , Pitman  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,   I.S. Sargsyan,   "Introduction to spectral theory: selfadjoint ordinary differential operators" , Amer. Math. Soc.  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Marchenko,   "Sturm–Liouville operators and applications" , Birkhäuser  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.M. Glazman,   "Direct methods of qualitative spectral analysis of singular differential operators" , Israel Program Sci. Transl.  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V. Hutson,   J. Pym,   "Applications of functional analysis and operator theory" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E.C. Titchmarsh,   "Eigenfunction expansions associated with second-order differential equations" , '''1''' , Clarendon Press  (1946)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  G.A. Mirzoev,   "Description of the self-adjoint extensions of quasi-regular operators generated by differential expressions with two terms"  ''Math. Notes'' , '''29''' :  2  (1981)  pp. 116–121  ''Mat. Zametki'' , '''29''' :  2  (1981)  pp. 225–233</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.M. Molchanov,   "On conditions for discreteness of the spectrum of a second-order differential equation"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 169–199  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  N. Levinson,   "Criteria for the limit-point case for second order linear differential operators"  ''Časopis Pěst. Mat. Fys.'' , '''74'''  (1949)  pp. 17–20</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R.S. Ismagilov,   "Conditions for self-adjointness of differential operators of higher order"  ''Soviet Math. Dokl.'' , '''3''' :  1  (1962)  pp. 279–283  ''Dokl. Akad. Nauk SSSR'' , '''142''' :  6  (1962)  pp. 1239–1242</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A.Ya. Povzner,   "On differential equations of Sturm–Liouville type on the half-line"  ''Mat. Sb.'' , '''23''' :  1  (1948)  pp. 3–52  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  W. Everitt,   "On the deficiency index problem for ordinary differential equations 1910–1976"  G. Berg (ed.)  et al. (ed.) , ''Differential Equations (Proc. Internat. Conf. Uppsala)'' , Almqvist &amp; Weksell  (1977)  pp. 62–81</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)  pp. Chapt. 10, §3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.M. Levitan,   I.S. Sargsyan,   "Sturm–Liouville and Dirac operators" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>